Chapter 3: Q. 83 (page 311)

Determine the local extrema of the function,
$f (x) = x^{3} e^{- x}, I = [0, \infty), J = (- \infty, \infty)$ .

Short Answer

Expert verified

The value

Step by step solution

Step 1. Given Information.

The function is,

$f (x) = x^{3} e^{- x}, I = [0, \infty), J = (- \infty, \infty)$ .

Step 2. The critical point.

For critical points we consider,

$f' (x) = 0 \frac{d}{d x} (x^{3} e^{- x}) = 0 x^{3} (- e^{- x}) + (e^{- x}) 3 x^{2} = 0 x^{2} e^{- x} (- x + 3) = 0$

Dividing both sides by $e^{- x}$ ,

$x^{2} (- x + 3) = 0$

Either $x = 0$

or, role="math" localid="1648585031235" $- x + 3 = 0 \Rightarrow x = 3$

Hence, $x = 0, 3$ .

Step 3. Values of I.

Now,

$\lim_{x \to 0} f (x) = \lim_{x \to 0} x^{3} e^{- x} = 0$

localid="1648585483256">limx→∞f(x)=limx→∞x3e-x=limx→∞x3ex[∞∞]=limx→∞6xex=limx→∞6ex=6∞=0

The graph of the function with limit $I = (0, \infty)$ is shown as,

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Determine the local extrema of the function,
$f (x) = x^{3} e^{- x}, I = [0, \infty), J = (- \infty, \infty)$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The critical point.

Step 3. Values of I.

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Determine the local extrema of the function,f(x)=x3e-x,I=[0,∞),J=(-∞,∞).

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. The critical point.

Step 3. Values of I.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Mechanics Maths

Probability and Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Determine the local extrema of the function,
$f (x) = x^{3} e^{- x}, I = [0, \infty), J = (- \infty, \infty)$ .